Singular values of some modular functions

Abstract

For an integer N greater than 5 and a triple ${\mathfrak{a}}=[a_{1},a_{2},a_{3}]$ of integers with the properties 0<a i ≤N/2 and a i ≠a j for i≠j, we consider a modular function $W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}$ for the modular group Γ 1(N), where ℘(z;L τ ) is the Weierstrass ℘-function relative to the lattice L τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples ${\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]$ and a pair of non-negative integers F=[m,n], we define a modular function $T_{{\mathfrak{A}},F}$ for the group Γ 0(N) as the trace of the product $W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}$ to the modular function field of Γ 0(N). In this article, we study the integrality of singular values of the functions $W_{\mathfrak{a}}$ and $T_{{\mathfrak{A}},F}$ by using their modular equations. We prove that the functions $T_{{\mathfrak{A}},F}$ for suitably chosen ${\mathfrak{A}}$ and F generate the modular function field of Γ 0(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values $T_{{\mathfrak{A}},F}(\tau)$ for suitably chosen ${\mathfrak{A}}$ and F generate ring class fields. Further, we study the class polynomial of $T_{{\mathfrak{A}},F}$ for Schertz N-system.